Octagon Algorithm for Border Wheel Squares (Part IV)

The Eight Node Way - 5×5 Squares

The previous section Part III introduced the Octagon A algorithm for the construction of 9×9 Wheel Squares. This section will cover Wheel Border squares a topic which was discussed in 2015 using 7×7 squares. The method of construction, then, required using some little difficult to follow math. I have found a the Octagon method is a better way of tackling this problem which doesn't use math at all but employs directed graphs to fill in the non spoke cells. Border squares are magic squares whose internal lower number magic squares are also magic. As stated above, border squares can be constructed using the Wheel algorithm but with the wheel portion first constructed but in reverse order from that listed in Part I. So for instance for 5×5 squares:

Thus, according to the discussion in Part I there are three wheels which can be produced: Wheel A, Wheel B and Wheel C:

which uses the accompaning 5×5 complementary table as a guide to pick the 17 wheel spoke numbers and 8 of the non spoke numbers:

To fill in the remainder of the numbers a new algorithm, Octagon E, is used starting with the initial node at the middle left of the octagon:

We start by adding the lowest spoke number on the complementary table to position (1,0), i.e., position (row, column), on the square. For example, in Wheel A, the first non spoke number, 7, goes into the cell corresponding to node 1. Using Octagon E one follows the arrows and deposits a number into the appropriate white cell of the square to generate the three squares. The difference between this algorithm and the previous ones is that the complements of the first four numbers no longer reside on the same row or column but reside across the octagon. Comparing the tuples of Octagon A from Part II with Octagon E, all tuples of A sum to 9 while those of E have sums of 8 and 10, respectively.

An alternative Octagon F, which has no edges superimposed onto the edges of the octagon, uses the following where node 1 is now at the bottom right of the octagon.

In addition, Border Fc is the way these border squares are normally depicted to show the magic properties of the internal and outer squares. The upshot is that both these algorithms along with the reversed wheel algorithm produce border squares, where the inner 3×3 squares have magic sums of 39 and the outer 5×5 squares have magic sums of 65.

Computer Coding for 5×5 Border Squares

Coding for these border squares is similar to those for the regular Octagon/Wheel squares, except on how the cells are filled. Programs 50-52 are the Octagon E series while programs 53-55 belong to the Octagon F series. Both programs go thru step by step procedure for filling of the non spoke section portion of each square. The table gives three programs each for both the Octagon E and F algorithms for the set of three squares each depicted above. The text coding is also available for both.

This completes the Octagon E and F method for 5×5 squares. Go to 7×7 Squares. Go back to homepage.